有限域上的三角和与和集问题

Arithmetic combinatorics is one of the most active subjects in nowadays mathematics. Due to the interplay of arithmetic combinatorics and analytic number theory, many new ideas and methods are invented through the outstanding contribution of J. Bourgain, W. T. Gowers, and T. Tao, and thus stimulate the development of analytic number theory. In this project, based on these new ideas and methods, we mainly are concerned with the following problems: 1) explicit estimates for the trigonometrical sums arised in arithmetic combinatorics, 2) the additive decomposition, counting and effective algorithms recognizing of sumsets in finite fields. It is expected to achieve substantive progress on the above issues through exploring the power of interdisciplinary integration.

算术组合是当今数学发展最为活跃的分支之一．在Bourgain，Gowers，Green，Tao等人的推动下，算术组合与解析数论交叉融合产生了许多新的思想与方法，推动了解析数论的发展．本项目将在这些新思想、新方法的基础上，对解析数论与算术组合中的一些问题进行研究，包括：1）算术组合中出现的三角和的显式估计；2）有限域中的和集分解问题，计数问题，以及和集的有效算法识别问题．我们期待通过学科交叉融合的力量，能够在上述问题上取得实质性进展．

本项目完成学术论文4篇．我们给出带有积性系数的Kloosterman和的非平凡上界估计．开展对剩余类环上Kloosterman和的任意正整数次幂矩的研究, 得到本质上完备的公式．给出有限素域中平移乘法子群上特征和的非平凡估计，并给出了两种类型的均值估计．完成对有限域中一般子集具有加性分解的必要条件的刻画．